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390077 A Decomposition Method Via Surrogate Modeling for Integrated Planning, Scheduling, and Dynamic Optimization of Sequential Batch Processes

Today’s process manufacturing enterprises call for new solutions so as to survive and grow in this highly competitive international marketplace [1]. To address the emerging challenges, enterprise-wide optimization (EWO) has attracted much interest [2-4]. An integrated optimization method proves to achieve a better performance than a conventional method which optimizes the subproblems independently in a sequential way [5, 6]. In recent decade, many integrated methods are presented, e.g. integrated scheduling and dynamic optimization [6-11], scheduling and control [12-14], etc. However, the integrated method usually suffers from severe computational complexity. Consequently, most existing integrated methods only focus on incorporating two decision layers. However, to exploit the full potential of EWO, more decision layers should be integrated and optimized collaboratively. Our goal is to devise efficient solution method, which can solve a complex problem integrating more than two decision layers.

In this work, we propose a novel framework to solve the integrated planning, scheduling, and dynamic optimization problem for batch processes. The integrated problem is formulated into a mixed-integer dynamic optimization problem (MIDO) [15], which contains binary decision variables associated with planning and scheduling decisions, and constraints of differential equations representing the dynamic models. The differential equations are then discretized by the collocation method [16], and they are transformed into a large set of nonlinear algebraic equations. The discretization procedure reformulates the integrated problem into a large-scale mixed-integer nonlinear program (MINLP). However, the formulated MINLP is too complex to be solved by simply invoking a commercial general-purpose MINLP solver [12].

To conquer the computational complexity, we develop efficient decomposition methods based on surrogate modeling. The decomposition methods exploit the special structure of the integrated problem to reduce the computational complexity. The integrated problem has a planning problem in the top level, a number of scheduling problems in the middle level, and a larger number of dynamic optimization problems in the bottom level. The problems in two adjacent levels are coupled by linking functions. Representing these linking functions by surrogate models breaks the coupling between the upper level problems and the lower level problems. Specifically, the lower level problems are solved to generate data points for building the surrogate models. The surrogate models are then substituted into the upper level problems replacing the detailed lower level problems. Next, the upper level problems are solved with the surrogate models. The determined variables of the upper level problems are passed to the lower level problems which are solved accordingly.

We develop two decomposition methods. The first method decomposes the dynamic optimization problems from the planning and scheduling problem. The large set of nonlinear constraints is replaced by the simple surrogate models representing the task operational recipes. The integrated MINLP is decomposed into a mixed-integer linear program (MILP) for the planning and scheduling problem and a set of separable dynamic optimization problems. The resulting MILP is much easier to solve than the complicated MINLP. Based on the first method, the second decomposition method further decomposes the planning and scheduling problem. The scheduling problem in a planning period is solved first to generate data points for building the surrogate models, which represent the production costs as a function of the production quantities. The generated surrogate models are used to solve the planning problem replacing the detailed scheduling problems.

The decomposition methods significantly reduce the computational complexity of the integrated problem. The computational efficiency is demonstrated in two case studies, including a large problem which the direct solution method fails to find a feasible solution after a long time period. The case studies also demonstrate the advantage of the integrated optimization methods in comparison to the conventional sequential method which neglects the interaction between the dynamic optimization problems and the scheduling problems.

**References**

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